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Audio compression using companders (integer implementation, A-Law, Mu-Law) for embedded micros with IIR fixed-radix averager

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# [Companders (a fixed point audio compression library)](https://deftio.github.io/companders/)

This repo is a simple audio compression library for microcontrollers using A-Law and Mu-Law (a type of compander).  The code uses fixed-radix (integer only) math and includes DC-offset correction with an IIR fixed-radix filter.

All the code in this repo is suitable for small microcontrollers such as 8-bit and 16-bit families (Arduino, 68HC, 8051) as well as 32-bit families such as ARM, RISC-V, etc.

As this is a fixed-point library (a type of integer math), floating point is not used or even emulated.  For more background on using fixed-point (or fixed-radix) math see this code library and documentation: [FR_Math](https://github.com/deftio/fr_math)

## Welcome

The accompanying companders.c contains a small set of functions written in C using only integer math for companding operations.  I developed this years ago for use in embedded integer math projects and this cleaned up version is made available for the public here.  It worked very well on M*CORE, 80x86 and ARM processors for small embedded systems.

Usage:

```C
#include "companders.h"   // no other dependencies or libraries are required

// .. in code then
int myCompandedValue = DIO_LinearToALaw(123); // convert the integer to its A-Law equivalent companded value

int unCompandedValue = DIO_ALawToLinear(myCompandedValue); // convert back to linear range. (with appropriate loss)
```

## About Companding

Companding is a special type of lossy compression in which linear format samples are converted to logarithmic representation to preserve the dynamic range of the original linear sample at the expense of uniform precision.  Companding is an old method and is free of patents (all have expired).

Theory of how companding works:
Suppose that we have a signed 16-bit linear sample in 2's complement math.  The range of this sample can vary from -32768 to +32767 in integer steps.  Now let's put the constraint that we only have 8 bits with which to represent the data, not the full 16 bits of the original linear number.  A simple way to preserve high order information would be to linearly truncate off the lower 8 bits giving us a signed number from -128 to +127.  We could make a mental note to keep track of the fact that we lost the lower 8 bits and so when we use this representation we multiply by 2^8 (256) to preserve the input range.
So:

```C
-128 * 256 = -32768
-127 * 256 = -32512
...
  -1 * 256 =   -256
   0 * 256 =      0
   1 * 256 =    256
   2 * 256 =    512
...
 126 * 256 =  32256 
 127 * 256 =  32512 
 ```

Notice that the steps between these linearly rounded off samples are quite large.  This truncated 8-bit representation would be very good at representing a linear quantity system such as a linear displacement transducer which moves through its whole range as part of normal operation (like a door), but terrible at logarithmic phenomena such as audio.  Audio information tends to be grouped around zero with occasional peaks of loudness.  So with linear quantization soft audio would be lost due to the large quantization steps at low volumes.  To address this, companders were developed for use in landline telephony for compressing audio data logarithmically instead of linearly.  Hence we use more bits for audio samples near zero and fewer for audio samples near the integer max extremes.

A-Law and its cousin Mu-Law are companders.  Rather than represent samples in linear steps, more bits are allocated to samples near zero while larger magnitude (positive or negative) samples are represented with proportionately larger interval sizes.

Differential encoding (taking the difference of neighboring samples) can greatly help with compressing data to where we "have more bits" as well, provided sample rates are fast enough.  Telephony typically operated with 8-bit non-differential (companded) samples at 8KHz.

In A-Law the following table (in bits) shows how a 13-bit signed linear integer is companded into A-Law:
(source is Sun Microsystems; a similar table is on Wikipedia or the G.711 specification).

|   Linear Input Code | Compressed Code |
|------------------------ | --------------- |
|   0000000wxyza   | 000wxyz         |
|   0000001wxyza   | 001wxyz         |
|   000001wxyzab   | 010wxyz         |
|   00001wxyzabc   | 011wxyz         |
|   0001wxyzabcd   | 100wxyz         |
|   001wxyzabcde   | 101wxyz         |
|   01wxyzabcdef   | 110wxyz         |
|   1wxyzabcdefg   | 111wxyz         |

### Mu-law (also mu-law or u-law)

Mu-Law is a similar compander to A-Law but used in American and Japanese telephony instead of European telephony.

Both algorithms aim to optimize the dynamic range of an audio signal by using logarithmic compression, but they differ in their compression characteristics and implementation. A-Law provides a slightly lower compression ratio, which offers better signal quality for signals with lower amplitude, whereas μ-Law provides higher compression, which can handle a wider range of input levels more efficiently but introduces more distortion for low-level signals. These standards were developed in the mid-20th century to improve the efficiency and quality of voice transmission over limited-bandwidth communication channels. A-Law was standardized by the International Telegraph and Telephone Consultative Committee (CCITT) and is detailed in the ITU-T G.711 recommendation, while μ-Law was standardized by the American National Standards Institute (ANSI).

## About this library

This free library supports A-Law, Mu-Law and IIR averages.  A-Law or Mu-Law are used for the companding while the IIR averagers can be used for embedded ADCs where the zero point is set loosely.  Since the companders are sensitive, and allocate more bits, to values near zero it's important to define a good zero.  For example, a microcontroller has a pin with an ADC which is fed a digitized audio signal.  The smallest value for the microcontroller is 0V = 0x00 and the 3.3V = 0x3ff for 10 effective bits of resolution.  A resistive analog divider is used to center the ADC near the half-input range or about 1.6V while the audio is capacitively coupled as shown here:

```text
 
             +3.3V
               |
               R
               |            
uC_ADC_Pin<-----------C---- audio_input_source
               |
               R
               |
              GND

```

However, cheap resistors have tolerances of 5 to 10%, so the setpoint voltage could easily be anywhere from 1.4 to 1.8V.  To address this, software should read the ADC before audio is coming in and determine the actual DC bias voltage set by the real-world resistors.  Then when audio is coming in, this value is subtracted from the ADC input to give a signed number which is fed to the LinearToALaw encoder.

If it is not easily possible to turn off the analog/audio source then the long-term DC average can be inferred by using one of the IIR functions included here with a long window length (perhaps 2000 samples if sampling at 8 KHz, or about 1/4 second).  These IIR functions are cheap to run in realtime even with reasonably high sample rates as they take little memory and are simple integer-math IIRs.

## About the Fixed Radix (FR) Math IIR averages

The (Infinite Impulse Response) IIR functions included here use fixed-radix math to represent fractional parts of an integer.  By providing a settable radix (number of bits devoted to fractional resolution), the programmer can make tradeoffs between resolution and dynamic range.  The larger the radix specified, the more bits that will be used in the fractional portion of the representation, which for smaller window sizes may not be necessary.  There are more comprehensive ways to deal with fractional representation in binary systems (floating point, bigNum/arbitrary length registers, separation of numerator/denominators, etc.) but these incur much larger compute/code/memory overhead.  The simple system used here avoids the need for testing for overflow/underflow which allows for low-footprint code/cpu/memory bandwidth.

To calculate how many bits of fractional part while preventing overflow use the following formulas:

```C
Nb = number_of_bits_in_use = ceiling(log2(highest_number_represented))

radix = 32-Nb(sample_resolution)-Nb(WindowLength)-2

//for example if you have an ADC generating counts from 0 to 1023 then the Nb(ADC-range) =
  Nb = ceiling(log2(1023)) = 10 // bits
  
//If you use a DC average window length of 2000 
  Nb = ceiling(log2(2000)) = 11 // bits
  
//so the max radix that should be specified is:
  radix = 32 - 10 - 11 - 2 = 32 - 23 = 9 // bits

```

With 9 bits in the radix, fractional precision of 512 units per integer (e.g. 1/512) is possible.  The "2" in the formula comes from reserving bits for the sign bit and the additional operation in the averager.

The function DIO_IIRavgFR() allows any integer length window size to be used, while the function DIO_IIRavgPower2FR() specifies window lengths in powers of two, but all calculations are based on shift operations which can be significantly faster on lower-end microprocessors.

## Embedded Systems

Now back to an embedded microcontroller example.  Let's say it has an ADC which maps the voltage on one pin from 0-3.3V into 0-4095 counts (12 bits).  We capacitively couple the input voltage and use the bias resistors to set the midpoint in the center of the range.  Let's say our resistors set the bias at 1.55V.  This equals our "zero point".  Below this point is negative from the incoming capacitively coupled audio and above this is positive.

Our "guess" as to the bias = both resistors are the same = (3.3V/2) = 1.65V = 1.65V * (4095 counts/3.3V) = 2048 counts

Resistor actual set bias "zero point" = 1.55V = 1.55V * (4095 counts/3.3V) = 1923 counts
We want this to be the "zero" we use for the companded A/D process.

To do this we start our ADC to periodically sample for some time before we start "recording" audio.  We will feed the ADC values into our IIR average to find the actual zero point.  Note that even when we decide not to "record" we can still run the A/D and IIR averager so it's adapting to the real zero point.

## C-like Pseudo code

```C
#include "companders.h"  // add src/ to your include path, or use the path directly

static volatile int32 gIIRavg= 2048; // global static var holds DC average as estimated by the IIR
static volatile int   gDoSomethingWithCompandedValue=0;

void processAudioSample() //interrupt handler -- runs at the sample rate
{
 short adcValue;
 char  aLawValue;
 
 //read the raw uncorrected sample (has some DC offset bias) 
 adcValue= inputPin.read_u16(); // read a 16 bit unsigned sample
 
 // this updates the IIR average every time the ADC returns a sample
 gIIRavg = DIO_IIRavgPower2FR(gIIRavg,11,adcValue,8);
 
 //now compute companded value with DC offset correction
 if (gDoSomethingWithCompandedValue)
 {
  aLawValue = DIO_LinearToALaw(adcValue-DIO_FR2I(gIIRavg));  // subtract off the offset and get a good A-Law value
  doSomethingWithALaw(aLawValue);  // store it to a buffer, send it the cloud ;) etc
 }
}

main()
{
 gDoSomethingWithCompandedValue=0;
 inputPin.attachInterrupt(&processAudioSample,8000); //attach interrupt handler, 8000 Hz sample rate
 
 // ... some point later in the code
 
 inputPin.start(); // start collecting audio.  Inside the interrupt handler we'll start estimating the DC bias
 
 // ... some time later  (when we actually want the audio) we set this flag
 gDoSomethingWithCompandedValue=1; //now the interrupt routine will call the doSomethingWithALaw() function
 
 // ... some time later
 gDoSomethingWithCompandedValue=0; //we're no longer collecting A-Law companded data, but we're
           // still running the IIR averager
 
}

```

The accompanying `examples/compandit.c` file is an example program demonstrating the convergence of the averager to a simulated DC offset value.

### Some Closing Comments

Finally, in some systems we can't turn off the audio input source — it may be hardwired to some sensor or mic, or perhaps the A/D center bias circuit (the 2 resistors) is always on when the audio is on.  In this case running the IIR with a long filter length all the time can remove the bias even when the audio is running.  For example, in an 8KHz sampling system an IIR length of 1024 is about 1/8 of a second or a cutoff freq well below 10Hz and costs almost nothing to run.

### Further Reading

* [TI App Note on A-Law and Mu-Law](https://www.ti.com/lit/an/spra163a/spra163a.pdf)
* [Wikipedia on Companding](https://en.wikipedia.org/wiki/Companding)
* [University of British Columbia Lab on Companding](https://people.ece.ubc.ca/edc/4550.jan2018/lab2.pdf)

### Code Coverage

Code coverage is achieved using gcov/lcov from the gcc test suite.  To see the code coverage:

```bash
make clean
make coverage
```

This builds with coverage flags, runs the tests, and generates an lcov summary.  The coverage report is also saved to `bin/coverage.info`.

### Project Structure

```
companders/
  src/                 - Library source (companders.h, companders.c)
  test/                - Test suite
  examples/
    arduino_compander/ - Arduino IDE example (.ino)
    platformio_compander/ - PlatformIO example (Arduino framework)
    espidf_compander/  - ESP-IDF native example
    compandit.c        - POSIX demo (built via makefile)
  .github/             - CI workflows
  library.json         - PlatformIO library manifest
  library.properties   - Arduino Library Manager manifest
  idf_component.yml    - ESP-IDF component manifest
  CMakeLists.txt       - ESP-IDF component build file
```

## Versions

* 1.0.7 09 May 2026  -- reorganized build tree (src/, test/, examples/), added Arduino example, updated CI, no api/code changes.
* 1.0.6 08 Aug 2024  -- added ci testing
* 1.0.5 30 May 2024  -- cleaned up release 
* 1.0.4 30 May 2024  -- add ulaw, updated docs, added full test 
* 1.0.3 23 Jan 2023  -- updated docs only (no api change)
* 1.0.3 28 Jul 2019  -- updated docs, ver example 
* 1.0.2 15 Jul 2016  -- updated README.md to markdown format.  updated license to be OSI compliant.  no code changes. some minor doc updates. Thanks to John R Strohm giving me the nudge to update the docs here. 
* 1.0.1  3 Sep 2012  -- original release in github

## License

See attached LICENSE.txt file  (OSI approved BSD 2 Clause)

Redistribution and use in source and binary forms, with or without
modification, are permitted provided that the following conditions are met:

* Redistributions of source code must retain the above copyright notice, this
  list of conditions and the following disclaimer.

* Redistributions in binary form must reproduce the above copyright notice,
  this list of conditions and the following disclaimer in the documentation
  and/or other materials provided with the distribution.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE
FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR
SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY,
OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

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